Existence of spike-layer solutions for singularly perturbed Choquard equations

Zhihui Zhang; Minbo Yang

doi:10.3934/dcds.2025099

Article Contents

2026, Volume 46: 258-286. Doi: 10.3934/dcds.2025099

This volume Previous Article Small and large data scattering for the dispersion-managed NLS Next Article Dynamics of the sine-Gordon equation on tadpole graphs

Existence of spike-layer solutions for singularly perturbed Choquard equations

Zhihui Zhang^1, , and
Minbo Yang^2,

1.
School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, China
2.
School of Mathematical Sciences, Zhejiang Normal University, Jinhua, Zhejiang, 321004, China

^*Corresponding author: Zhihui Zhang
^*Corresponding author: Zhihui Zhang

Received: January 2025

Revised: June 2025

Early access: June 27, 2025

Published online: June 27, 2025

The first author was partially supported by NSFC(12271028) and Minbo Yang was partially supported by NSFC(11971436, 12011530199) and ZJNSF(LZ22A010001).

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Abstract

In this paper, we consider the following singularly perturbed nonlinear elliptic problem with Dirichlet boundary condition,

$ \begin{equation} \nonumber \left\lbrace \begin{aligned} &-\varepsilon^{2}\Delta u+u = \varepsilon^{\mu-N}\left( \displaystyle{\int}_{\Omega}\frac{\left|u(y)\right| ^{p}}{\left| x-y\right| ^{\mu}}dy\right) \left|u\right| ^{p-2}u,\; u>0&\text {in}\; \Omega,\\ &u\left( x\right) = 0&\text {on}\; \partial \Omega. \end{aligned} \right. \end{equation} $

Under certain conditions on $ p,N $, and $ \mu $, we first obtain a Liouville type result for the Choquard equation on the half space. Then, through an energy estimate, we prove that the mountain pass solution $ u_{\varepsilon} $ of the problem is a "spike-layer" solution with a sharp spike located in the most-centered part of $ \Omega $ as $ \varepsilon $ tends to 0.

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Mathematics Subject Classification: Primary: 35A15, 35B25; Secondary: 35J25.

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